High-order Div- and Quasi Curl-Conforming Basis Functions ... · (EFIE) is presented. In contrast...

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Abstract— A new high-order Calderón multiplicative

preconditioner (HO-CMP) for the electric field integral equation (EFIE) is presented. In contrast to previous CMPs, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of high-order quasi curl-conforming basis functions. Like its predecessors, the HO-CMP can be seamlessly integrated into existing EFIE codes. Numerical results demonstrate that the linear systems of equations obtained using the proposed HO-CMP converge rapidly, regardless of the mesh density and of the order of the current expansion.

Index Terms— Integral equations, EFIE, high-order basis functions, preconditioning.

I. INTRODUCTION

LECTRIC field integral equation (EFIE) solvers find widespread use in the analysis of time-harmonic scattering

from perfect electrically conducting (PEC) surfaces [1]. This paper presents a new Calderón multiplicative preconditioner (CMP) for the EFIE which, unlike its predecessors, allows for high-order surface representations and current expansions.

The numerical solution of the EFIE requires the discretization of the scatterer’s surface in terms of a mesh of planar or curvilinear triangles or quadrangles, and of its current distribution, by means of N vector basis functions. Discretization of the EFIE leads to a dense N N× system of linear equations in the basis functions’ expansion coefficients. The computational cost of iteratively solving this system scales as 2( )iterNO N ; here 2N is the complexity of multiplying the system matrix with a trial solution vector and

iterN is the number of iterations required for convergence to a

Manuscript received May 17, 2010. This work was supported by the

National Science Foundation under Grant DMS 0713771, by the AFOSR STTR Grant "Multiscale Computational Methods for Study of Electromagnetic Compatibility Phenomena" (FA9550-10-1-0180), by the Sandia Grant "Development of Calderón Multiplicative Preconditioners with Method of Moments Algorithms", by the KAUST Grant 399813, and an equipment grant from IBM.

F. Valdés and E. Michielssen are with the Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI USA. (e-mail: [email protected])

F. P. Andriulli is with the Politecnio di Torino, Torino 10100, Italy. K. Cools is with the Department of Information Technology (INTEC),

Ghent University, B-9000 Ghent, Belgium. Color versions of one or more of the figures in this paper are available

online at http://ieeexplore.ieee.org.

prescribed residual. There exist many “fast methods” that reduce the complexity of a matrix-vector multiplication from

2( )O N to g( )loO N N [2-5]. Often iterN scales with the condition number of the system matrix, with small condition numbers guaranteeing fast convergence. Unfortunately, the condition number of the EFIE system matrix grows rapidly as the mesh discretization density increases [6]. As a result, the cost of solving the EFIE for structures with subwavelength geometric features often is prohibitively high.

Techniques for preconditioning the EFIE by leveraging Calderón identities have become quite popular in recent years [7-13]. In essence, these techniques exploit the self-regularizing property of the EFIE operator, viz. the fact that the square of the EFIE operator is a compact perturbation of the identity, to produce well-conditioned system matrices even when the mesh includes subwavelength geometric features. Unfortunately, few Calderón preconditioners developed to date are easily integrated into existing codes. The CMP technique proposed in [9] is one of them. The CMP uses two separate discretizations of the EFIE, one in terms of standard Rao-Wilton-Glisson (RWG) basis functions [14], and the other in terms of Buffa-Christiansen (BC) basis functions [15]. The latter are div- and quasi curl-conforming, and geometrically nearly orthogonal to the RWG functions. The effectiveness of the RWG-BC combination in the construction of the CMP stems from the fact that the RWG and BC functions are linked by a well-conditioned Gram matrix and guarantee the annihilation of the square of the discretized hypersingular component of the EFIE operator. We note that Chen and Wilton proposed basis functions similar to the BC ones in the context of analyzing scattering from penetrable objects [16]. Both the BC and Chen-Wilton basis functions are of zeroth-order and designed for use in conjunction with RWG basis functions.

In the last decade, EFIE solvers that use high-order representations of the surface and/or the current density have become increasingly popular. A high-fidelity representation of the surface can be achieved using a high-order parametric mapping from a reference cell to the scatterer surface, usually in the form of curvilinear patches (as opposed to flat ones). Among the many high-order basis functions for representing surface current densities, those proposed by Graglia-Wilton-Peterson (GWP( )p ) [17], which comprise of products of scalar polynomials (complete up to order p ) and RWG basis

High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative

Preconditioning of the EFIE

Felipe Valdés, Francesco P. Andriulli, Kristof Cools, and Eric Michielssen, Fellow, IEEE

E


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functions, are very popular. For a given solution accuracy, high-order EFIE solvers have been shown to be more CPU and memory efficient than their zeroth-order counterparts [18]. That said, they still suffer from ill-conditioning when applied to structures with subwavelength geometric features. To allow for a high-order CMP, a high-order extension of the BC functions is called for. Jan et al. [19] already presented an extension of the BC basis functions on curvilinear triangular patches; unfortunately their method does not extend to high-order current representations.

This paper presents a true high-order BC extension, viz. a set of high-order div- and quasi curl-conforming functions that, when used in conjunction with the GWP( )p functions, exhibits the aforementioned properties of the BC-RWG pair. The proposed basis functions are constructed as orthogonal projections of the range of the EFIE operator onto div-conforming GWP( )p s defined on a barycentrically refined mesh; preliminary insights into the construction of these basis functions were presented in [20]. Using these basis functions, a high-order CMP (HO-CMP) is implemented and its effectiveness demonstrated via a suite of numerical examples.

II. CALDERÓN PRECONDITIONED EFIE AND ITS DISCRETIZATION

This section describes the CMP EFIE idea. Section II-A describes the standard EFIE and its classical discretization. Section II-B describes the Calderón-preconditioned EFIE along with its CMP discretization.

A. Non-preconditioned EFIE solver

Consider a closed, simply connected PEC surface S residing in a homogeneous medium with permittivity ε and permeability µ . The (scaled) current density J on S induced by the incident time-harmonic electric field incE satisfies the EFIE [21] ˆ[ ] inc

r= − ×J n ET (1)

where [ ] [] ][ s h= +JJ JT T T (2)

with

'

ˆ ( ') ']'4

[ik

s r

S

eikds

π

−

= ×−∫r r

J rJr r

nT (3)

and

'

ˆ ·[ ] ''

( ')4

ik

h r S

S

id

es

kπ

−− ′ ′= × ∇ ∇−∫r r

n J rJr r

T . (4)

Here, k ω µ= ε and ˆ

rn is the outward pointing unit vector normal to S at r ; ω is the angular frequency. A time dependence i te ω− ( 1i = − ) is assumed and suppressed. The subscripts “s” and “h” stand for “singular” (vector potential) and “hyper-singular” (scalar potential), respectively. To numerically solve (1), S is approximated by a mesh Sδ of planar or curvilinear triangles with minimum edge size δ , and J is expressed as

1

(( ) )j

N

jj

I=

≈∑ rJ r f (5)

where jI , , ,1j N= … are expansion coefficients of J in terms of a set of the div-conforming basis functions

{ }( ), 1, ,j NF j == …rf . Throughout this paper it is assumed that F is the set of pth-order interpolatory Graglia-Wilton-Peterson functions, i.e.

GWP( )F p= [17]. These functions interpolate at 1p + and ( 1)p p+ nodes along each of the EN edges and on each of

the PN patches in Sδ , respectively; the total number of GWP( )p functions therefore is ( 1) ( 1)E PN p N p p N= + + + ; note that RWG GWP(0)= [17]. GWP( )p functions that interpolate at a node internal to a patch or on an edge henceforth will be referred to as patch and edge functions, respectively. For later use we note the Euler identity for a simply connected surface 2V PENN N− + = , (6)

where VN is the number of vertices in Sδ . Substitution of expansion (5) into (1), and testing the

resulting equation with curl-conforming functions in { }ˆ ( ), 1, , | ( )i ir i Fn NF × = … ∈= frfn r yields the N N×

linear system of equations F F=T I V , (7)

where

, , [ ]( ) ˆi j r i jF = ×T n f fT , (8)

( ) j jI=I , (9)

and

( ˆ ,) ˆ inci rF i r= ×− ×V n f n E . (10)

Here ), · (( )S

dsδ

= ∫ b ra b a r denotes the inner product between to vector functions a and b on Sδ .

When analyzing electromagnetic phenomena involving electrically large and/or complex structures, i.e., when N is large, (7) cannot be solved directly and iterative solvers are called for. The computational cost of solving (7) iteratively is proportional to the cost of multiplying the impedance matrix

FT by a trial solution vector and the number of iterations iterN required to reach a desired residual error; iterN typically is proportional to FT ’s condition number, viz. the ratio of F sT ’s largest and smallest singular values. Unfortunately, the singular values of the operator T comprise two branches, one accumulating at zero, and the other at infinity [6]. Thus the condition number of FT grows without bound as J is increasingly well-approximated, i.e. as 0δ → and/or p → ∞ . When this happens the number of iterations required for convergence often is prohibitively high.

B. Calderón preconditioned EFIE solver

A well-conditioned EFIE can be obtained by leveraging T’s self-regularizing property expressed by the Calderón identity [6,9-10],

2 2[ ] [ ]4

= − +JJ JT K (11)

with

'ˆ

[ ]( )'

( ') '4

ikr

S

se

dπ

−

′= ∇−

× ×∫r r

Jr

rr

nJ rK . (12)


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The operator K is compact on smooth surfaces: its singular values accumulate at zero and the same holds true for 2

K [6,22]. It follows that the operator 21/ 4− +K has a bounded spectrum with singular values accumulating at 1/ 4− . Eqn. (11) implies that the Calderón-preconditioned EFIE 2 ˆ[[ ] ]inc

r= − ×n EJT T (13)

may be amenable to stable discretization regardless of the mesh density or basis function order.

Unfortunately, the discretization of [ ]2[ ] [ ]=J JT T T is by no means trivial. The literature abounds with techniques for discretizing 2 2 2[ ] [ ] [ ] [ ][ ]s s h h s h= + ++J J J JJT T T T T T T (14)

that separately handle the first three terms in the above expansion, explicitly leaving out the fourth as 2 0h ≡T [7-8]. However, the implementation of these techniques into existing codes is quite intrusive. The CMP proposed in [9] does not suffer from this drawback. The CMP approximates 2[ ]JT as the product of two impedance matrices

�FT and FT with

{ }( ), 1, ,jF j N= = …f rɶɶ , separated by a Gram matrix that accounts for the possible lack of (bi-)orthogonality between the functions in Fɶ and nF . In other words, the CMP matrix equation reads CMP CMPI = VT (15)

where � �

CMP 1

;F nF F F−=T T G T , (16)

� �( )CMP 1

; FF nF F

−=V T G V , (17)

and

� ,;) ˆ ,( i j r i jnF F

= ×G n f fɶ (18)

is the matrix of overlap integrals of functions in Fɶ and nF . Eqn. (15) does not require the decomposition of matrix elements in

�FT and FT into their singular (vector potential)

and hypersingular (scalar potential) components, simplifying its implementation. That said, (15) only will be well-conditioned if C1. the functions in Fɶ and F are div-conforming; C2. the matrix �;nF F

G is well-conditioned; this ensures the rapid iterative solution of � ( ); FnF F

G y = T x for trial solution vectors x while solving (15); this requirement precludes the choice GWP( )F F p= =ɶ as such leads to a singular Gram matrix;

C3. the sets Fɶ and F ensure the cancellation of 2[ ]h JT upon discretization, i.e.

� �

1

; ,,0

nF F h Fh F

− =T G T (19)

where

, , ˆ( [) , ]i jh F r i h j= ×T n f fT . (20)

If (19) is not satisfied, the desirable spectral properties of 2T

will not be inherited by � �

1

;F nF F F−T G T .

The above criteria are satisfied by the sets RWGF = and BCF =ɶ , the set of (zeroth-order) div- and quasi curl-

conforming Buffa-Christiansen basis functions, used by all CMP implementations reported to date [9-10,12-13,19].

III. ZEROTH-ORDER QUASI CURL-CONFORMING BASIS

FUNCTIONS

This section reviews the construction of the BC basis functions and their main properties [9-10,15].

Just as RWGF = , the set BCF =ɶ contains EN N= basis functions. Contrary to the current of the RWG function nf , which crosses edge n (Fig. 1(a)), that of the BC function nfɶ flows along edge n (Fig. 1(c)). Consider the barycentrically refined mesh Sδ , obtained by adding the three medians to each triangle of the original mesh Sδ . Each BC basis function is a linear combination of div-conforming RWGs defined on Sδ [9-10,15]. Even though BC functions are strictly div-conforming, they also are quasi curl-conforming in that they resemble curl-conforming RWGs in nF (Fig. 1(b)). This renders the Gram matrix in (18) (with RWGF = and

BCF =ɶ ) well-conditioned. That is, the sets RWGF = and BCF =ɶ fulfill conditions C1 and C2 above. To show that

these sets also satisfy condition C3, consider the space ( ) ( )Span SpansolF F⊂ spanned by “div-conforming

solenoidal RWG” functions

{ }( ), 1, ,sol solj

solj NF = …= f r (21)

with 1solVN N= − ; the sol

jf are charge-free and could, for example, be “loop” functions describing current flowing around all but one of the vertices in Sδ (Fig. 2(a)) [23-24]. The set solF can be complemented by a set nonsolF such that

( ) ( ) ( )Span Span Spansol nonsolF F F= ⊕ . The set nonsolF contains “div-conforming non-solenoidal RWG” functions

{ }( ), 1, ,nonsol nons nonj

oo s llF j N= …= rf (22)

with ( 1) 1nonsolV PN N NN = − − = − ; the nonsol

jf all produce charge and could, for example, be “star” functions describing current flowing out of all but one patch in Sδ (Fig. 2(b)) [23-24]. Similarly, consider the space ( ) ( )Span SpansolF F⊂ɶ ɶ spanned by “div-conforming solenoidal BC” functions

{ }( ), 1, , nonsolsol sol soljF N Nj == =…f rɶɶ ɶ . (23)

The dimensionality of ( )Span solFɶ equals that of ( )Span nonsolF ; indeed, it can be verified that an appropriate

linear combination of the BC functions associated with the three edges of a patch in Sδ describes a divergence-free current circulating the patch (Fig. 2(c)) [10]. The set solFɶ can be complemented by a set nonsolFɶ such that

( ) ( ) ( )Span Span Spanso nl onsolF F F⊕=ɶ ɶ ɶ . The set nonsolFɶ contains “div-conforming non-solenoidal BC” functions

{ }( ), 1, ,nonsol nonsolj

nonsol soljF N N= … == f r ɶɶɶ . (24)

Again, the dimensionality of ( )Span nonsolFɶ equals that of ( )Span solF (Fig. 2(d)) [10].

Next, assume that the matrices ,h FT , �,h FT , and �;nF F

G , are not constructed using the sets RWGF = and BCF =ɶ , but instead from sol nonsolF F∪ and nonsol solF F∪ɶ ɶ with functions in the left and right subset labeled 1 through 1VN − and VN through N , respectively; note the reverse order of the “sol” and “nonsol” superscripts for functions in RWGF = and

BCF =ɶ . It is clear from (20) and (4) that the entries , ,( )h F i jT and � ,,

( )i jh FT vanish when the source function is solenoidal or

the test function is irrotational, which implies


4

�

�,,

,,

an, dnonsol

nonsolh

h FF Fh

h F

= =

TT T

T

00 0

0 0 0 (25)

The blocks in these matrices have dimensions

( ) (1 ( 1) 1 ( 1)

1 ( 1) 1 ( 1

)

)( ) ( )V V V P

P V P P

N N N

N N

N

N N

− × − − × −−

× − − × −

. (26)

Since an irrotational function can be written as the surface gradient of a scalar function φ , and a solenoidal function can be written as the surface curl of a scalar function ψ , the inner product of two such functions can be expressed as

( ) ( )ˆ( ) · ( )S

S r S dsφ ψ∇ ×∇∫ r n r (27)

which can be transformed by partial integration into

( )ˆ( ) · ( ) 0S r

S

Sr n r dsφ ψ∇ ×∇ =∫ . (28)

Therefore, the Gram matrix �;nF FG has the form

�;nF F

=

B 0

C DG (29)

and so does its inverse

�

1

;nF F

− =

0

'D'

B'G

C. (30)

From (25) and (30), it is clear that � �

1,, ; h Fh F nF F

− =T G T 0 . The fact that the dimension of the solenoidal subspace of the RWG basis functions equals that of the non-solenoidal subspace of the BC basis functions (and vice-versa), is essential for the CMP technique to work, as it ensures the cancellation of

2[ ]h JT upon discretization.

IV. HIGH-ORDER QUASI CURL-CONFORMING BASIS FUNCTIONS

In this section, a set of div- and quasi curl-conforming high-order extensions of the BC functions is proposed. In the construction of (15)-(18), for any 1p ≥ , the new set Fɶ is meant to be used alongside RWG RWG h

sol nonsolh o hooF F F F= ∪ = ∪ ∪ . (31)

Here hoF complements the set of RWG functions such that

( ) ( ) ( )Span GWP( ) Spa S an RWG p n hop F= ⊕ . The sets

{ }, ( ), 1, ,sol sol solho jho hoF j N= …= rf (32)

and

{ }, ( ), 1, ,nonsol nonsol nonsolho jo hh oF j N= …= f r (33)

span the solenoidal and non-solenoidal subspaces of

( )Span hoF , respectively. Likewise, Fɶ will be constructed as

C BCB hnoo

nsol solo hohF FF F∪ ∪= ∪=ɶ ɶɶ ɶ , (34)

with sets

{ }, ,( ), 1,nonsol nonsoh

l nonsolho j ho oF j N…= =f rɶ ɶɶ (35)

and

{ }, ( ), 1, ,sol solh

solho j hoo jF N= = …f r ɶɶ ɶ (36)

judiciously chosen such that system (15) is well-conditioned. Throughout this section, notation introduced previously for

spaces and sets applicable to F will be reused and extended

for all spaces and functions derived from the barycentrically refined mesh Sδ by adding bars on top of symbols. That is,

RWG RWG hsol nonsol

h o hooF F F F= ∪ = ∪∪ , (37)

where the sets

{ }, ( ), 1, ,ssol soolho ho j

lhoF j N= …= rf (38)

and

{ }, ( ), 1, ,snonsol nonsoolh ho ho j

loF j N= …= rf (39)

span the solenoidal and non-solenoidal subspaces of ( )Span hoF , respectively; GWP( )p and RWG denote

GWP( )p and RWG basis functions defined on Sδ ; etc. To guarantee that system (15) has a low condition number,

sets Fɶ and F must satisfy the above conditions C1 through C3. To ensure functions in Fɶ are div-conforming, they will be constructed as linear combinations of the div-conforming functions in F . Bases for the high-order solenoidal and non-solenoidal subspaces ( )Span sol

hoFɶ and ( )Span hononsolFɶ will be

built separately. To arrive at a well-conditioned Gram matrix �;nF F

G , functions in olhosFɶ ( no

honsolFɶ ) will be constructed so as to

“resemble” those in noho

nsolnF ( olhosnF ). To ensure the

cancellation � �

1

; ,,0

nF F h Fh F

− =T G T , the cardinality of solhoFɶ

( noho

nsolFɶ ) will be matched to that of noho

nsolF ( solhoF ) i.e.,

sol nonsolho hoN N=ɶ and nonsol sol

ho hoN N=ɶ . Section IV-A details the Helmholtz decomposition of the

spaces hoF , hoF , and hoFɶ . Section IV-B explains how to construct the hoFɶ basis functions such that the Gram matrix

�;nF FG is well-conditioned.

A. Helmholtz decomposition

As described in [25], bases for ( )Span hl

osoF and ( )Span ho

nonsolF can be constructed by separating ( )Span hoF into edge and (internal-to-) patch subspaces. One way of constructing these subspaces is as follows: 1. For each patch in Sδ ;

a. Define A as the matrix that maps all ( 1)p p+ patch GWP( )p functions (columns of A ) onto their charges (divergence) at points in the patch (rows of A )

b. Perform a singular value decomposition (SVD) on A : T=A UΣV .

c. The last ( 1) / 2p p− columns of TV are associated with zero singular values, and they describe patch solenoidal functions (Fig. 3(a)).

d. All other columns of TV describe patch non-solenoidal functions (Fig. 3(b)).

2. For each edge in Sδ ; a. Define A as the matrix that maps 1p + edge

GWP( )p and the 2 ( 1)p p+ overlapping patch GWP( )p functions (columns of A ) onto their charges at points in the patches (rows of A ).

b. Define Tj jβ= + ∑B A u v , where β is a non-zero

constant, and ju and jv are the singular vectors associated with the patch solenoidal functions identified in step 1 for the two patches that share the edge. Note that the patch solenoidal functions are now associated with singular values equal to β .

c. Perform a SVD on B : T=B UΣV .


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d. The last p columns of TV are associated with zero singular values, and they describe edge solenoidal functions (Fig. 3(c)).

To summarize, the set lhosoF contains:

(i) ( 1) / 2Pp p N− high-order patch functions (ii) and EpN high-order edge functions. Likewise, no

honsolF contains ( )( 2)( 1) 2 / 2Pp p N+ + − high-

order patch functions. The cardinalities of lhosoF and no

honsolF are

therefore

( 1)

2solho P E

p pN N pN

−= + , (40)

and ( )( 2) / 2( 1) 2nonsol

ho PNpN p+ + −= . (41)

Of course ( 1)sol nonsolho ho E P EN N pN p p N N N= + + = −+ .

Once the high-order solenoidal and non-solenoidal functions have been obtained as described above, they can be (separately) linearly combined to form a more convenient basis for ( )Span sol

hoF and ( )Span nonsolhoF , respectively. A

partial local orthogonalization can be performed as follows: 1. For each edge in Sδ , orthogonalize the solenoidal

functions associated with it. 2. For each patch in Sδ , separately orthogonalize the

solenoidal and non-solenoidal functions. After this partial orthogonalization has been performed, all functions in nonsol

hoF are orthogonal to one another, but not necessarily orthogonal to any or all functions in sol

hoF . Furthermore, among the functions in sol

hoF , only those which are patch based are orthogonal to one another, but not necessarily orthogonal to any or all of the edge solenoidal functions.

A full local orthogonalization can also be performed. The difference with respect to the previous one being that now patch-based solenoidal and non-solenoidal functions are orthogonalized altogether. Hence all functions in nonsol

hoF are orthogonal to one another, and also orthogonal to all patch based functions, but not necessarily to any or all edge based functions in sol

hoF . For future use, we define the matrix L (of size sol

hoN N× ) that expresses functions in sol

hoF as linear combinations of functions in GWP( )p , i.e. its j th column contains the coefficients obtained for ,

lho jsof after the orthogonalization

process. Similarly, the matrix S (of size nonsolhoN N× )

expresses functions in nonsolhoF as linear combinations of

functions in GWP( )p . Next, consider the barycentric refinement Sδ of Sδ . As Sδ is simply connected, so is Sδ ; hence Sδ has

2 6EE PNNN = + edges, V V E PNN NN= + + vertices, and 6P PN N= patches. The total number of high-order solenoidal

and non-solenoidal functions is

( ) ( )2

( 1)

2

3 3 1 2 1

solho P E

P E

p pN N

p p Np

pN

N

−=

= + +

+

+ + (42)

( )

( 2)( 1)1

2

3 ( 2)( 1) 2

nonsolho P

P

p pN N

p p N

+ + −

= + +

−

= (43)

Matrices L and S can be obtained just as described before, operating on the functions in GWP( )p .

Each function in solhoFɶ is built as a linear combination of

functions in solhoF , i.e.

,, ,1

( ) ( ), 1, ,solN

sol sol solho j ho k

k

nonsolk j hoj Np

=

= = …∑ rf frɶ ’ (44)

Likewise, each function in nonsolhoFɶ is built as a linear

combination of functions in nonsolhoF , i.e.

, ,1

,( ) ( ) 1,, ,nonsolN

nonsol nonsol nonsol solkho j ho k

kj hop j N

=

= = …∑f r f rɶ . (45)

Note that the cardinality of solhoFɶ ( no

honsolFɶ ) matches that of

noho

nsolF ( solhoF ), thereby ensuring the cancellation

� �

1,, ; h Fh F nF F

− =T G T 0 .

B. Well-conditioning of the Gram matrix

Due to the way the sets F and Fɶ are defined, the Gram matrix �;nF F

G can be decomposed into four blocks

�

�

�

RWG;BC RWG;

;;BC ;

ho

ho hoho

n n F

nF FnF nF F

=

G GG

G G, (46)

where RWG;BCnG , �RWG; hon FG , ;BChonFG , and

�; hohonF FG are

matrices of size E EN N× , ( )E EN N N× − , ( )E EN NN− × , and ( ) ( )E EN N N N×− − respectively. The block RWG;BCnG is nothing but the Gram matrix encountered in the zeroth-order case, and it is of course well-conditioned [9-10]. That said, in order for the block

�; hohonF FG to be well-conditioned the

expansion coefficients ,solk jp in (44), and ,

nonsolk jp in (45) need to

be chosen appropriately. Clearly, if we insist that

, ,ˆ ( ) ,) 1,( ,sol nonsol nonsol

ho j r ho j hoj N≈ × = …f n f rrɶ (47)

and

, ,( ) ˆ ( ) , 1, ,nonsol sol solho j r ho j hoj N× = …≈ n f rf rɶ , (48)

then the entries of �; hohonF F

G will be approximately those of

;ho hoF FG . This suggests that the condition number of the former matrix should be similar to that of the latter.

To achieve the resemblance in (47), ,sol

ho jfɶ is chosen to be the orthogonal projection of ,

ˆ nonsolr ho j×n f onto ( )Span sol

hoF , i.e.,

, , , 0, 1, ,, ˆsol noho i ho j ho j

nsol sol solr hoi N−× ∀ …= =f n f fɶ . (49)

Substituting (44) into (49) yields a system of linear equations for the expansion coefficients ,

solk jp :

, ,,1

, ,ˆ, , 1, , ,

solNsol sol sol sol nonsol solk j hoho i ho k i r j hoho

k

i Np=

×= ∀ = …∑ f f f n f .(50)

Eqn. (50) can be expressed in matrix form as

; ;

1sol sol sol nonsol

ho ho ho ho

solho F F F nF

−P G= G (51)

where the Gram matrices ;sol sol

ho hoF FG and

;sol nonsolho hoF nF

G are

; , , ,,( )sol sol

ho ho

sol soli j ho i hoF jF

=G f f (52)

, , ,;( ,) ˆsol nonsol

ho ho

sol nonsoli j ho i hrF nF o j×=G f n f , (53)


6

and , ,( )sol solho k j k jp=P . Similarly, enforcing (48) yields the

following equation for the expansion coefficients ,nonsolk jp :

;

1

;nonsol nonsol nonsol solho ho ho ho

nonsolho F F F nF

−P G= G (54)

with

,; , ,) ,( nonsol nonsolho ho

nonsol nonsoli j ho i ho jF F

=G f f , (55)

, , ,;ˆ,( )nonsol sol

ho ho

nonsol soli j rF n jF ho i ho×=G f n f , (56)

and , ,( )nonsol nonsolho k j k jp=P .

As an example on how these two orthogonal projections perform, consider the div-conforming patch (edge) solenoidal function ,

solho jf depicted in Fig. 3(a) (Fig. 3(c)). Its curl-

conforming counterpart ,ˆ sol

r ho j× fn is shown in Fig. 4(a) (Fig. 4(c)). The orthogonal projection of the latter is the div-conforming patch (edge) non-solenoidal function ,

nonsolho jfɶ ,

depicted in Fig. 4(b) (Fig. 4(d)). Similarly, consider the patch non-solenoidal function ,

nonsolho jf depicted in Fig. 3(b). Its curl-

conforming counterpart ,ˆ nonsol

r ho j×n f is shown in Fig. 4(e). The orthogonal projection of the latter is the div-conforming patch solenoidal function ,

solho jfɶ depicted in Fig. 4(f). Note that the

support of ,sol

ho jfɶ is a couple of “barycentric patches bigger” than the support of ,

ˆ nonsolr ho j×n f . This “extra space” is required

as a return path for the current described by ,sol

ho jfɶ to provide a charge-free approximation of ,

ˆ nonsolr ho j×n f .

Since the functions in solhoFɶ ( nonsol

hoFɶ ) are built to resemble those in nonsol

honF ( solhonF ), the condition numbers of

�; hohonF FG

and �;nF FG are expected to depend on the way the functions in

solhoF and nonsol

hoF are obtained. Indeed, if functions in solhoF and

nonsolhoF are not orthogonalized in any way described at the end

of section IV-A, the condition numbers of �; hohonF F

G and �;nF FG

grow without bound with p . As it will be shown later in section VI, partial and full local orthogonalization of the functions in sol

hoF and nonsolhoF reduce the aforementioned

growth on the condition numbers to a minimum. An ideal scenario would be one in which the functions in sol

hoF and nonsol

hoF are built as one orthogonal set of functions, such that

;ho hoF FG equals the identity. Hence the matrix �; hohonF F

G would be as close as it can be to the identity matrix. Of course, such orthogonalization cannot be performed “locally” therefore it is far from being practical due to its computational cost.

V. IMPLEMENTATION OF THE HO-CMP

This section provides details on the construction of the basis functions in Fɶ and their use in the HO-CMP. First, explicit expressions for the matrices sol

hoP and nonsolhoP are given in

terms of Gram matrices and basis transformations. With these matrices, expressions for

�FT , �;nF F

G , and FT are given. Finally, issues relating to computational cost are discussed.

The evaluation of solhoP in (51) requires the computation of

two Gram matrices: ;sol sol

ho hoF FG and

;sol nonsolho honFF

G . Since each function in sol

hoF is a linear combination of functions in GWP( )p , the Gram matrix

;sol solho hoF F

G in (52) can be obtained as the product

;

T;sol sol

ho ho F FF FG = L G L . (57)

Similarly, the Gram matrix ;sol nonsol

ho honFFG in (53) can be

computed as

;

T;sol nonsol

ho ho F FF nnF=G L G R S , (58)

where R is the matrix (of size N N× ) that expresses functions in GWP( )p as linear combinations of functions in GWP( )p .

The evaluation of nonsolhoP in (54) involves the computation

of ;nonsol nonsol

ho hoF FG and

;nonsol solho honFF

G . The former can be obtained as the product

;

T;nonsol nonsol

ho ho F FF FG = S G S. (59)

Finally, ;nonsol sol

ho honFFG can be computed as

;

T;nonsol sol

ho ho FF F Fn n=G S G R L . (60)

Substitution of the above expressions into eqns. (51) and (54) yields

( ) ( )1T T; ;

solho F F F nF

−=P L G L L G R S , (61)

( ) ( )1T T; ;

nonsolho F F F nF

−=P S G S S G R L . (62)

All matrices on the right hand side of (61) and (62) are sparse and can be multiplied by a vector in ( )O N operations. Note that the inversion of matrix ( )T

;F FL G L need not be performed explicitly, instead its operation on any vector x can be obtained by solving the linear system ( )T

;F FL G L y = x iteratively. Of course, orthogonalization of the functions in sol

hoF as described in the previous section makes the matrix well-conditioned. Thus, the evaluation of (61) has an overall computational cost that scales as ( )O N . Similar considerations apply to the evaluation of (62), with the exception that if the functions in nonsol

hoF are orthogonalized, then ( )T

;F FS G S is nothing but the identity, therefore no system need to be solved.

The implementation of the HO-CMP follows the same structure of the zeroth-order CMP (see [9]), which makes use of matrices zoP and zoR , that express functions in BC and RWG as linear combinations of functions in RWG , respectively. The matrix zoP encountered in the zeroth-order CMP is extended here to P defined as

zo

solho

nonsolho

=

P 0 0

0 0 P

0 P 0

P , (63)

where solhoP and nonsol

hoP are given in (61) and (62) respectively. Explicit expressions for the entries of zoP can be found in [9]. The matrix zoR encountered in the zeroth-order CMP is replaced here by the matrixR , defined earlier in this section. Using P , the matrix

�FT in (15) can be discretized as follows:

�

T TF F FF

=T P H T H P , (64)

where FH is the matrix that expresses functions in F as linear combinations of functions in GWP( )p . Similarly, FT in (15) can be discretized as T T

FF F F=T R T R HH , (65)

where FH is the matrix that expresses functions in F as linear combinations of functions in GWP( )p . Finally, with �;nF F

G being discretized as �

T T;; F nF F FnF F

=G R G HH P , (66)

the evaluation of (16) and (17) can now be carried out.


7

The computational cost of solving (15) is that of multiplying the matrix CMPT times the number of iterations required to reach a prescribed residual error. Evaluation of a vector times

CMPT involves multiplying first by FT as in (65), then by the inverse of

;nF FG ɶ as in (66), and finally by

FTɶ as in (64). As

mentioned previously, the cost of multiplying R , solhoP , and

nonsolhoP by a vector scales as ( )O N . Thus, the cost of

multiplying P (and therefore ;nF F

G ɶ ) by a vector also scales as ( )O N . Provided that

;nF FG ɶ is well-conditioned, and it is,

then its inverse can be multiplied by a vector using just a few (i.e., (1)O ) iterations of an iterative solver like the generalized minimal residual (GMRES) [26] or the transpose-free quasiminimal residual (TFQMR) [27]. Using the multilevel fast multipole method [3], the cost of multiplying FT by a vector scales as ( )TC O N+ where TC is the cost of multiplying FT by a vector. Indeed, even though the dimension of FT is greater that that of FT by a factor of 6, the additional degrees of freedom introduced by the barycentric mesh do not change the number of multipoles required for field expansion compared to that used when multiplying by FT . Therefore, the cost of multiplying FT increases only by an additive linear term. The fact that the number of iterations required for the HO-CMP to converge is much smaller than that of the standard EFIE justifies the use of the former scheme.

VI. NUMERICAL RESULTS

This section presents several examples that demonstrate the effectiveness of the basis functions presented in this paper and its performance in the HO-CMP. The results emphasize the main advantage of using a HO-CMP: high-order accuracy in the solutions, without compromising the number of iterations needed for convergence. The results presented here are obtained using a parallel EFIE MoM solver, which uses the proposed HO-CMP or a standard diagonal preconditioner. This solver uses a TFQMR-based iterative method [27] to solve the EFIE MoM systems.

A. High-order accuracy

The first two examples demonstrate the convergence of the radar cross section (RCS) as the order of the basis functions in the HO-CMP is increased. Each example comprises a smooth PEC object: a sphere of radius 1 m., and a star-shaped object whose surface is parameterized as

2 2, sin (2( ) 1 )cos ( ).5r θ φ θ φ= + m., both illuminated by a 30 MHz., x -polarized plane wave traveling in the z direction. Fig. 5(a) (Fig. 6(a)) shows the bistatic RCS of the PEC sphere (star-shaped object) when computed with basis functions of orders 0,1,2,3p = . Fig. 5(b) (Fig 6(b)) shows the relative error of the computed RCS of the PEC sphere (star-shaped object) with respect to Mie series (4th-order) solution. In these examples, the geometric models consist of 32 patches for the sphere and 102 patches for the star-shaped object. Each patch is obtained by means of an exact mapping from a reference patch onto the surface of the object. The evaluation of basis functions on curvilinear patches requires the computation of a Jacobian function, which requires additional computation time

when compared to flat patches [17]. The overhead introduced by the evaluation of the Jacobian is more than compensated however by the reduction in the number of patches required to accurately decribe the sphere surface.

B. Condition number

The following three examples illustrate the behavior of the condition numbers of the non-preconditioned EFIE and HO-CMP system matrices as the surface current expansion is increasingly well-approximated, i.e. as 0δ → and/or p → ∞ . Table I shows the condition numbers of �;nF F

G , FT , and CMPT , obtained with several mesh discretizations of the PEC

sphere of Fig. 5(a) using basis functions of orders 1,2,3,4p = . Similarly, Tables II and III show the same data

for the star-shaped object of Fig. 6(a) and a PEC cube with sidelength of 1 m., respectively. These results show that for a fixed order p , the condition numbers of �;nF F

G and CMPT remain bounded as the mesh density is increased, whereas the condition number of FT does not.

By virtue of the Calderón identity in (11), the operator 2T

is spectrally equivalent to the identity operator. Hence the condition number of CMPT depends on how well the sets F and Fɶ can discretize the identity operator, i.e. the Gram matrix �;nF F

G . As mentioned in section IV-B, the growth in the condition number of �;nF F

G (and therefore of CMPT ) with p is related to the way in which the functions in sol

hoF and nonsol

hoF are obtained. Table IV shows the condition numbers of �;nF F

G and CMPT for three different ways of obtaining these sets, and for orders 1,2,3,4p = . As expected, full local orthogonalization of the functions in sol

hoF and solhoF result in

lower condition numbers for the matrices �;nF FG and CMPT

that are more stable with respect to p when compared to partial local orthogonalization. Also, as conjectured at the end of section IV-B, a global orthogonalization of the functions in

solhoF and sol

hoF yields �;nF FG and CMPT matrices with condition

numbers that are almost independent of p .

C. Speed of convergence

The three examples in this section compare the speed of convergence of the diagonally-preconditioned EFIE and HO-CMP when solved iteratively. Figs. 7(a-e) show the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems obtained by discretizing the diagonally-preconditioned EFIE and HO-CMP with basis functions of orders 1,2,3,4,5p = . The geometry is a PEC sphere of radius 1 m. Similarly, Figs. 8(a-e) show the same data for a PEC cube with sidelength of 1 m. In both examples, the excitation is a 30 MHz., x -polarized plane wave traveling in the z direction, and the prescribed accuracy (relative residual error) for the TFQMR solver is 510− . As dictated by the condition number of CMPT , the number of iterations required for the HO-CMP to reach the prescribed accuracy does not grow as the discretization density is increased. In contrast, the diagonally-preconditioned EFIE requires an increasing number of iterations as the mesh becomes denser. Moreover, this behavior worsens as the order p of the basis functions is


8

increased, severely penalizing the efficiency and accuracy of high-order basis functions. Next, the diagonally-preconditioned EFIE and HO-CMP are used to analyze scattering from a printed monopole antenna similar to the one presented in [28]. The antenna geometry and mesh are shown in Fig. 9(a). Note that the dielectric substrate has not been considered here. The antenna is fed with a voltage delta-gap. The divergence of the electric current, i.e. the (scaled) charge distribution on the surface of the antenna is plotted in Fig. 9(b). The current distribution in this example was obtained using the HO-CMP, with basis functions of order 1p = and a frequency of 3.55 GHz. The radiation pattern of the antenna is plotted in Fig. 9(c) for two different frequencies: 3.55 and 5.5 GHz. Finally, Fig. 9(d) shows the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems stemming from the diagonally-preconditioned EFIE and HO-CMP with basis functions of orders 0,1p = . The last example involves a model of the Airbus A380 shown in Fig. 10(a). The surface of the aircraft is discretized using second-order curvilinear patches, allowing the use of (relatively) large patches on smooth surfaces (wings and main body), and small patches near fine geometric features (engines and wing tips). The airplane is illuminated by a y -polarized plane wave traveling in the x direction. Fig. 10(b) shows the bistatic RCS obtained for four different frequencies, ranging from 1.5 to 30 MHz. Fig. 10(c) and 10(d) show the divergence of the current density induced on the surface of the aircraft, at frequencies of 6 MHz and 30 MHz, respectively. Note that at 30 MHz the high-order basis functions allow for the use of less than 5 patches per wavelength on the wings and main body of the aircraft. Finally, Fig. 10(e) shows the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems obtained by discretizing the diagonally-preconditioned EFIE and HO-CMP with basis functions of orders 1,2,3p = . In this case, the excitation frequency is 6 MHz. Similarly, Fig. 10(f) shows the residual error versus iteration count achieved by a TFQMR solver for an excitation frequency of 30 MHz. Using basis functions of order 0p = , it took 30 minutes and 16852 iterations for the diagonally preconditioned EFIE to converge to a prescribed relative residual error of 410− . For the HO-CMP it took 11 minutes and 485 iterations. Using basis functions of order 1p = , the diagonally preconditioned EFIE could only reach a relative residual error of 31.8 10−× after 8.6 hours and 100000 iterations. For the HO-CMP it took 1.2 hours and 383 iterations to reach the prescribed relative residual error of 410− .

VII. CONCLUSION

In this paper, the CMP technique is extended to high-order by building a set of high-order div- and quasi curl-conforming basis extensions of the BC basis functions used by all CMP implementations reported to date. Numerical results demonstrate fast convergence rates of the HO-CMP, regardless of the mesh density and the order of the basis functions used. The HO-CMP presented here can be used in

the presence of open surfaces with minor modifications. In addition, the basis functions presented here can also be used in high-order Calderón preconditioned formulations for analyzing scattering from penetrable objects.

REFERENCES [1] A. J. Poggio, and E. K. Miller, “Integral equation solutions of three-

dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. New York: Pergamon, 1973.

[2] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propagat., vol. 45, no. 3, pp. 533-543, Mar. 1997.

[3] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast-multipole algorithm for solving electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propagat., vol. 45, no. 10, pp. 1488-1493, Oct. 1997.

[4] E. Bleszynski, M. Bleszynski, and T. Jaroszewic, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31 (1996), pp. 1225–1151.

[5] A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14 (1993), pp. 1368–1393.

[6] J.-C. Nedéléc, Acoustic and Electromagnetic Equations. New York: Springer-Verlag, 2000.

[7] H. Contopanagos, B. Dembart, M. Epton, J.J. Ottusch, V. Rokhlin, J.L. Visher, and S.M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propagat., vol.50, no.12, pp. 1824-1830, Dec 2002.

[8] R. J. Adams, and N.J. Champagne, “A numerical implementation of a modified form of the electric field Integral equation,” IEEE Trans. Antennas Propagat., vol.52, no.9, pp. 2262-2266, Sept. 2004.

[9] F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A Multiplicative Calderón Preconditioner for the Electric Field Integral Equation,” IEEE Trans. Antennas Propagat., vol.56, no.8, pp.2398-2412, Aug. 2008.

[10] K. Cools, F.P. Andriulli, F. Olyslager, and E. Michielssen, “Time-Domain Calderón Identities and their Application to the Integral Equations analysis of Scattering by PEC objects Part I: Preconditioning,” IEEE Trans. Antennas Propagat., vol. 57, pp. 2352-2364, Aug 2009.

[11] F. P. Andriulli, and E. Michielssen, “A Regularized Combined Field Integral Equation for Scattering from 2-D Perfect Electrically Conducting Objects,” IEEE Trans. Antennas Propagat., vol. 55, pp. 2522-2529, Sept. 2007.

[12] F. Valdés, F.P. Andriulli, H. Bagci, and E. Michielssen, “On the discretization of single source integral equations for analyzing scattering from homogeneous penetrable objects,” Antennas and Propagation Society International Symposium, 2008. AP-S 2008. IEEE , vol., no., pp.1-4, 5-11 July 2008.

[13] M. B. Stephanson, and Jin-Fa Lee, “Preconditioned Electric Field Integral Equation Using Calderón Identities and Dual Loop/Star Basis Functions,” IEEE Trans. Antennas Propagat., vol. 57, pp. 1274-1279, Apr. 2009.

[14] S. M. Rao, D. R. Wilton, and A W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409-418, May 1982.

[15] A. Buffa and S. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comp., vol. 76, pp. 1743-1769, 2007.

[16] Q. Chen and D. R. Wilton, “Electromagnetic scattering by three-dimensional arbitrary complex material/conducting bodies,” in Proc. IEEE AP-S Symp., vol. 2, 1990, pp. 590-593.

[17] R.D. Graglia, D.R. Wilton, and A.F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propagat., vol.45, no.3, pp.329-342, Mar 1997.

[18] K. C. Donepudi, J. M. Jin, S. Velamoarambil, J. M. Song, and W. C. Chew, “A higher-order parallelized multilevel fast multipole algorithm for 3D scattering,” IEEE Trans. Antennas Propagat., vol. 49, pp. 1078-1078, July 2001.

[19] S. Yan, J-M. Jin, and Z. Nie, “Implementation of the Calderón Multiplicative Preconditioner for the EFIE Solution with Curvilinear Triangular Patches,” presented in Antennas and Propagation Society International Symposium, 2009. AP-S 2009. IEEE , June 2009.


9

[20] F. Valdés, F.P. Andriulli, K. Cools, and E. Michielssen, “High-Order Quasi-Curl Conforming Functions for Multiplicative Calderón Preconditioning of the EFIE,” presented in Antennas and Propagation Society International Symposium, 2009. AP-S 2009. IEEE , June 2009.

[21] G. C. Hsiao and R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propagat., vol. 53, pp. 3316-3323, Oct. 2005.

[22] R. Kress, Linear Integral Equations, 2nd edition, Springer, New York, 1999.

[23] J.S. Zhao, and W.C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1635-1645, 2000.

[24] W. Wu, A.W. Glisson, and D. Kajfez, “A study of two numerical solutions procedures for the electric field integral equation at low frequency,” Appl. Comput. Electromagn. Soc. J., vol. 10, pp. 69-80, 1995.

[25] R.A. Wildman, and D.S. Weile, “An accurate broad-band method of moments using higher order basis functions and tree-loop decomposition,” IEEE Trans. Antennas Propagat., vol.52, no.11, pp. 3005-3011, Nov. 2004.

[26] Y. Saad, and M.H. Schultz, “GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems,” SIAM J. Sci. Stat. Comput., 3, vol. 7, 1986, 856-869.

[27] R. W. Freund, “A transpose-free quiasi-minimal residual algorithm for non-Hermitian linear systems,” SIAM J. Sci. Comput. Volume 14, Issue 2, pp. 470-482, 1993.

[28] M. A. Antoniades, and G. V. Eleftheriades, “A Broadband Dual-Mode Monopole Antenna Using NRI-TL Metamaterial Loading,” IEEE Antennas and Wireless Propagation Letters., vol.8, pp. 258-261, May 2009.

Felipe Valdés received his B.S. degree in electrical engineering from the Pontificia Universidad Católica de Chile, Santiago, Chile, in 2004. Since August 2006 he has been a Research Assistant at the Radiation Laboratory, University of Michigan at Ann Arbor, where he is currently working toward the Ph.D. degree in electrical engineering. His main research interest is in computational electromagnetics, with focus on preconditioning and single source integral equations.

Mr. Valdés was the recipient of a Fulbright Doctoral Fellowship in 2006-2010.

Francesco P. Andriulli (S’05, M’09) received the Laurea degree in electrical engineering from the Politecnico di Torino, Italy, in 2004, the M.S. degree in electrical engineering and computer science from the University of Illinois at Chicago in 2004, and the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor in 2008. Since 2008 he has been a Research Associate with the Politecnico di Torino. His research interests are in computational

electromagnetics with focus on preconditioning and fast solution of frequency and time domain integral equations, integral equation theory, hierarchical techniques, and single source integral equations. Dr. Andriulli was awarded the University of Michigan International Student Fellowship and the University of Michigan Horace H. Rackham Predoctoral Fellowship. He was the recipient of the best student paper award at the 2007 URSI North American Radio Science Meeting. He received the first place prize of the student paper context of the 2008 IEEE Antennas and Propagation Society International Symposium, where he authored and coauthored other two finalist papers. He was the recipient of the 2009 RMTG Award for junior researchers and was awarded a URSI Young Scientist Award at the 2010 International Symposium on Electromagnetic Theory.

Kristof Cools was born in Merksplas, Belgium, in 1981. He received the M.S degree in physical engineering from Ghent University, Belgium, in 2004. His master's dissertation dealt with the full wave simulation of meta-materials using the low frequency multilevel fast multipole method. In August 2004, he joined the Electromagnetics Group of the Department of Information Technology (INTEC) at Ghent University. Since then, he has been working towards his PhD degree

under the advisership of Prof. Femke Olyslager (Ghent University) and Prof. Eric Michielssen (University of Michigan). His research focuses on the spectral properties of the boundary integral operators of electromagnetics.

Eric Michielssen (M’95–SM’99–F’02) received his M.S. in Electrical Engineering (Summa Cum Laude) from the Katholieke Universiteit Leuven (KUL, Belgium) in 1987, and his Ph.D. in Electrical Engineering from the University of Illinois at Urbana-Champaign (UIUC) in 1992. He joined the faculty of the UIUC Department of Electrical and Computer Engineering in 1993, reaching the rank of Full Professor in 2002. In 2005, he joined the University of Michigan as

Professor of Electrical Engineering and Computer Science. Since 2009, he directs the University of Michigan Computational Science Certificate Program. Eric Michielssen received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. Furthermore, he was the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. In addition, he was named 1999 URSI United States National Committee Henry G. Booker Fellow and selected as the recipient of the 1999 URSI Koga Gold Medal. He also was awarded the UIUC's 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony Scholar. He is a Fellow of the IEEE (elected 2002) and a member of URSI Commission B. Eric Michielssen served as the Technical Chairman of the 1997 Applied Computational Electromagnetics Society (ACES) Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterey, CA), and served on the ACES Board of Directors (1998-2001 and 2002-2003) and as ACES Vice-President (1998-2001). From 1997 to 1999, he was as an Associate Editor for Radio Science, and from 1998 to 2008 he served as Associate Editor for the IEEE Transactions on Antennas and Propagation. Eric Michielssen authored or co-authored over one 160 journal papers and book chapters and over 280 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His research focuses on the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices.


10

Fig. 1. RWG and BC functions defined for edge n in Sδ . Functions are

plotted on top of Sδ . (a) Div-conforming RWG, nf . (b) Curl-conforming

RWG, ˆr n×n f . (c) Div-conforming BC, nfɶ . (d) Curl-conforming BC,

ˆr n×n fɶ .

Fig. 2. Div-conforming RWG and BC solenoidal and non-solenoidal

functions defined in Sδ . Note that functions are plotted on top of Sδ . (a)

Div-conforming RWG solenoidal function solnf , describing current flowing

around vertex n in Sδ . (b) Div-conforming RWG non-solenoidal function nonsol

nf , describing current flowing out of patch n in Sδ . (c) Div-

conforming BC solenoidal function solnfɶ , describing current flowing around

patch n in Sδ . (d) Div-conforming BC non-solenoidal function nonsolnfɶ ,

describing current flowing out of vertex n in Sδ .


11

Fig. 3. Div-conforming hoF solenoidal and non-solenoidal functions defined

in Sδ . Note that functions are plotted on top of Sδ . (a) Div-conforming hoF

patch solenoidal function ,l

ho jsof , its support (shaded area) is limited to a patch

in Sδ . (b) Div-conforming hoF patch non-solenoidal function ,no

h jl

onsof , its

support (shaded area) is limited to a patch in Sδ . (c) Div-conforming hoF

edge solenoidal function ,l

ho jsof , its support (shaded area) include the two

patches sharing the edge in Sδ .

TABLE I

CONDITION NUMBERS OF ;nF F

G ɶ , FT , AND CMPT FOR THREE DIFFERENT

MESH DISCRETIZATIONS OF A PEC SPHERE

p PN N ;nF FG ɶ FT CMPT

32 160 11.77 335.87 21.18 102 510 13.4 2318.48 25.63 1 224 1120 14.35 5542.08 32.95 32 336 45.06 2942.35 63.31 102 1071 68.08 21298.59 98.67 2 224 2352 59.97 47297.28 88.26 32 576 62.59 25681.78 202.45 102 1836 72.94 189540.53 233.97 3 224 4032 78.02 417912.93 265.26 32 880 156.68 201766.16 571.16 102 2805 183.08 1604161.36 705.08 4 224 6160 192.28 3394907.21 740.86

TABLE II



MESH DISCRETIZATIONS OF A PEC STAR-SHAPED OBJECT


32 160 13.24 239.07 23.56 102 510 14.49 1167.83 26.5 1 224 1120 14.06 4013.27 28.53 32 336 51.38 2005.19 78.68 102 1071 91.44 11268.25 125.03 2 224 2352 70.19 36240.98 109.66 32 576 82.46 17867.56 262.67 102 1836 88.62 102686.45 279.92 3 224 4032 83.03 325654.23 262.08 32 880 179.14 145715.16 700.56 102 2805 226.84 905399.7 926.65 4 224 6160 198.94 2689593.01 735.11

TABLE III



MESH DISCRETIZATIONS OF A PEC CUBE


24 120 14.94 658.14 26.63 154 770 12.41 6719.43 35.38 1 240 1200 12.21 10411.1 42.66 24 252 76.75 5266.6 118.11 154 1617 73.03 59766.95 115.66 2 240 2520 59.98 97635.17 96.78 24 432 69.48 50812.32 221.55 154 2772 69.5 582413.73 315.89 3 240 4320 71.17 841216.68 338.87 24 660 172.38 325648.19 674.44 154 4235 165.76 3910371 881.96 4 240 6600 168.5 6382122.93 921.35


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Fig. 4. Div- and quasi curl-conforming functions in hoFɶ , approximating those in honF . Note that functions are plotted on top of Sδ . (a) ,ˆ sol

r ho j×n f , i.e.

curl-conforming counterpart of the patch solenoidal function ,l

ho isof depicted in Fig.3 (a). (b) Div-conforming patch non-solenoidal function ,

nonsolho jfɶ

approximating ,ˆ solr ho j×n f . (c) ,ˆ sol

r ho j×n f , i.e. curl-conforming counterpart of the edge solenoidal function ,l

ho isof depicted in Fig.3 (c). (d) Div-conforming

edge non-solenoidal function ,nonsol

ho jfɶ approximating ,ˆ solr ho j×n f . (e) ,ˆ nonsol

r ho j×n f , i.e. curl-conforming counterpart of the patch non-solenoidal function

,no

h jl

onsof depicted in Fig.3 (b). (f) Div-conforming patch solenoidal function ,

solho jfɶ approximating ,ˆ nonsol

r ho j×n f .


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Fig. 5. Bistatic RCS of a PEC sphere of radius 1 m. illuminated by a 30

MHz x -polarized plane wave traveling in the z direction. The surface of

the sphere is modeled with 32 curvilinear patches. The current density is

modeled with basis functions of orders 0,1,2,3p = . The number of

unknowns ranges from 48 ( 0p = ) to 576 ( 3p = ): (a) Bistatic RCS in the x-

z plane. (b) Relative error in the RCS with respect to Mie series solution.

Fig. 6. Bistatic RCS of a PEC star-shaped object illuminated by a 30 MHz

x -polarized plane wave traveling in the z direction. The surface of the

object is modeled with 102 curvilinear patches. The current density is

modeled with basis functions of orders 0,1,2,3p = . The number of

unknowns ranges from 153 ( 0p = ) to 1836 ( 3p = ). (a) Bistatic RCS in the

x-z plane. (b) Relative error in the RCS with respect to the solution obtained

using basis functions of order 4p = .

TABLE IV


G ɶ , AND CMPT FOR THREE DIFFERENT HELMHOLTZ DECOMPOSITION STRATEGIES

Partial local orthogonalization Full local orthogonalization Full global orthogonalization

p N ;nF FG ɶ CMPT ;nF F

G ɶ CMPT ;nF FG ɶ CMPT

1 160 29.9 21.18 29.9 20.99 2.51 3.14 2 336 73.1 63.31 55.5 48.15 2.54 3.17 3 576 152.3 202.45 126.5 134.97 2.79 3.29 4 880 220.2 571.16 177.4 248.44 2.94 3.47


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Fig. 7. Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines) for the case of a PEC sphere of radius 1 m.,

illuminated by a 30 MHz., x -polarized plane wave traveling in the z direction. Four different discretizations are used, ranging from 32 to 810

curvilinear elements. Results are shown for several orders of the basis functions: (a) order 1; (b) order 2; (c) order 3; (d) order 4; (e) order 5.


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Fig. 8. Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines) for the case of a PEC cube of side 1 m., illuminated

by a 30 MHz., x -polarized plane wave traveling in the z direction. Four different discretizations are used, ranging from 24 to 918 elements. Results are

shown for several orders of the basis functions: (a) order 1; (b) order 2; (c) order 3; (d) order 4; (e) order 5.


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Fig. 9. Monopole antenna excited with a voltage delta-gap. (a) Mesh and dimensions of the antenna. (b) Divergence of the current density induced on

the antenna, for a frequency of 3.55 GHz. (c) Radiation pattern in the x-y plane for two different frequencies. (d) Residual history of diagonally-

preconditioned EFIE (dashed lines) and HO-CMP (solid lines), for a frequency of 5.5 GHz for orders 0,1p = .


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Fig. 10. Airbus A380 model illuminated by y -polarized plane wave traveling in the x direction. (a) Mesh and dimensions of the aircraft; second order

curvilinear patches are used to discretize the surface. (b) Bistatic RCS in the x-y plane for four different frequencies. (c) Divergence of the current

density induced on the aircraft, for a frequency of 6 MHz. (d) Divergence of the current density induced on the aircraft, for a frequency of 30 MHz. (e)

Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines), for a frequency of 6 MHz for orders 1,2,3p = . (f)

Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines), for a frequency of 30 MHz for orders 0,1,2p = .

High-order Div- and Quasi Curl-Conforming Basis Functions ... · (EFIE) is presented. In contrast...

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